We devise a new undecimated wavelet thresholding for denoising images corrupted by additive Gaussian white noise. The first key point of our approach is the use of a linearly parameterized pointwise thresholding function. The second key point consists in optimizing the parameters globally by minimizing Stein's unbiased MSE estimate (SURE) directly in the image-domain, and not separately in the wavelet subbands.